# Properties

 Label 20.0.23808916007...2224.1 Degree $20$ Signature $[0, 10]$ Discriminant $2^{10}\cdot 17^{4}\cdot 2297^{4}$ Root discriminant $11.72$ Ramified primes $2, 17, 2297$ Class number $1$ Class group Trivial Galois Group 20T964

# Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 9, 0, 32, 0, 62, 0, 70, 0, 50, 0, 28, 0, 19, 0, 12, 0, 5, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 5*x^18 + 12*x^16 + 19*x^14 + 28*x^12 + 50*x^10 + 70*x^8 + 62*x^6 + 32*x^4 + 9*x^2 + 1)
gp: K = bnfinit(x^20 + 5*x^18 + 12*x^16 + 19*x^14 + 28*x^12 + 50*x^10 + 70*x^8 + 62*x^6 + 32*x^4 + 9*x^2 + 1, 1)

## Normalizeddefining polynomial

$$x^{20}$$ $$\mathstrut +\mathstrut 5 x^{18}$$ $$\mathstrut +\mathstrut 12 x^{16}$$ $$\mathstrut +\mathstrut 19 x^{14}$$ $$\mathstrut +\mathstrut 28 x^{12}$$ $$\mathstrut +\mathstrut 50 x^{10}$$ $$\mathstrut +\mathstrut 70 x^{8}$$ $$\mathstrut +\mathstrut 62 x^{6}$$ $$\mathstrut +\mathstrut 32 x^{4}$$ $$\mathstrut +\mathstrut 9 x^{2}$$ $$\mathstrut +\mathstrut 1$$

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

## Invariants

 Degree: $20$ magma: Degree(K); sage: K.degree() gp: poldegree(K.pol) Signature: $[0, 10]$ magma: Signature(K); sage: K.signature() gp: K.sign Discriminant: $$2380891600778249012224=2^{10}\cdot 17^{4}\cdot 2297^{4}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $11.72$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $2, 17, 2297$ magma: PrimeDivisors(Discriminant(K)); sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{578} a^{18} + \frac{57}{578} a^{16} - \frac{203}{578} a^{14} - \frac{133}{578} a^{12} - \frac{1}{2} a^{11} + \frac{24}{289} a^{10} - \frac{55}{578} a^{8} - \frac{1}{2} a^{7} - \frac{189}{578} a^{6} + \frac{30}{289} a^{4} + \frac{131}{289} a^{2} - \frac{1}{2} a + \frac{25}{289}$, $\frac{1}{578} a^{19} + \frac{57}{578} a^{17} + \frac{43}{289} a^{15} - \frac{1}{2} a^{14} + \frac{78}{289} a^{13} - \frac{1}{2} a^{12} + \frac{24}{289} a^{11} + \frac{117}{289} a^{9} - \frac{1}{2} a^{8} + \frac{50}{289} a^{7} - \frac{1}{2} a^{6} - \frac{229}{578} a^{5} - \frac{1}{2} a^{4} - \frac{27}{578} a^{3} - \frac{1}{2} a^{2} - \frac{239}{578} a - \frac{1}{2}$

magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk

## Class group and class number

Trivial Abelian group, order $1$

magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp

## Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
 Rank: $9$ magma: UnitRank(K); sage: UK.rank() gp: K.fu Torsion generator: $$-1$$ (order $2$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); sage: UK.torsion_generator() gp: K.tu[2] Fundamental units: $$\frac{160}{289} a^{18} + \frac{450}{289} a^{16} + \frac{466}{289} a^{14} + \frac{106}{289} a^{12} + \frac{455}{289} a^{10} + \frac{1893}{289} a^{8} - \frac{762}{289} a^{6} - \frac{3983}{289} a^{4} - \frac{3742}{289} a^{2} - \frac{959}{289}$$,  $$a^{19} + 5 a^{17} + 12 a^{15} + 19 a^{13} + 28 a^{11} + 50 a^{9} + 70 a^{7} + 62 a^{5} + 32 a^{3} + 9 a$$,  $$\frac{1342}{289} a^{19} + \frac{6267}{289} a^{17} + \frac{13973}{289} a^{15} + \frac{20635}{289} a^{13} + \frac{30314}{289} a^{11} + \frac{56529}{289} a^{9} + \frac{74377}{289} a^{7} + \frac{56822}{289} a^{5} + \frac{22433}{289} a^{3} + \frac{4098}{289} a$$,  $$\frac{670}{289} a^{19} - \frac{1669}{578} a^{18} + \frac{6153}{578} a^{17} - \frac{3783}{289} a^{16} + \frac{6756}{289} a^{15} - \frac{8187}{289} a^{14} + \frac{19745}{578} a^{13} - \frac{23673}{578} a^{12} + \frac{14531}{289} a^{11} - \frac{17514}{289} a^{10} + \frac{27308}{289} a^{9} - \frac{33144}{289} a^{8} + \frac{70709}{578} a^{7} - \frac{84535}{578} a^{6} + \frac{26328}{289} a^{5} - \frac{61125}{578} a^{4} + \frac{9943}{289} a^{3} - \frac{22563}{578} a^{2} + \frac{1421}{289} a - \frac{3397}{578}$$,  $$\frac{317}{578} a^{19} + \frac{805}{578} a^{18} + \frac{509}{289} a^{17} + \frac{1990}{289} a^{16} + \frac{626}{289} a^{15} + \frac{9407}{578} a^{14} + \frac{611}{578} a^{13} + \frac{7302}{289} a^{12} + \frac{672}{289} a^{11} + \frac{10650}{289} a^{10} + \frac{2409}{289} a^{9} + \frac{38379}{578} a^{8} + \frac{777}{578} a^{7} + \frac{26667}{289} a^{6} - \frac{7279}{578} a^{5} + \frac{22705}{289} a^{4} - \frac{7403}{578} a^{3} + \frac{10374}{289} a^{2} - \frac{1779}{578} a + \frac{1918}{289}$$,  $$\frac{1185}{578} a^{19} - \frac{1291}{578} a^{18} + \frac{5699}{578} a^{17} - \frac{3125}{289} a^{16} + \frac{13187}{578} a^{15} - \frac{14211}{578} a^{14} + \frac{10065}{289} a^{13} - \frac{10530}{289} a^{12} + \frac{29425}{578} a^{11} - \frac{30467}{578} a^{10} + \frac{26802}{289} a^{9} - \frac{56733}{578} a^{8} + \frac{36419}{289} a^{7} - \frac{76791}{578} a^{6} + \frac{30059}{289} a^{5} - \frac{29193}{289} a^{4} + \frac{25805}{578} a^{3} - \frac{10749}{289} a^{2} + \frac{4629}{578} a - \frac{2993}{578}$$,  $$\frac{615}{578} a^{19} - \frac{543}{578} a^{18} + \frac{3265}{578} a^{17} - \frac{2051}{578} a^{16} + \frac{8095}{578} a^{15} - \frac{3637}{578} a^{14} + \frac{12997}{578} a^{13} - \frac{2183}{289} a^{12} + \frac{18827}{578} a^{11} - \frac{3495}{289} a^{10} + \frac{33223}{578} a^{9} - \frac{7465}{289} a^{8} + \frac{24103}{289} a^{7} - \frac{14129}{578} a^{6} + \frac{21629}{289} a^{5} - \frac{2418}{289} a^{4} + \frac{10916}{289} a^{3} + \frac{789}{578} a^{2} + \frac{4451}{578} a + \frac{8}{289}$$,  $$\frac{615}{578} a^{19} + \frac{543}{578} a^{18} + \frac{3265}{578} a^{17} + \frac{2051}{578} a^{16} + \frac{8095}{578} a^{15} + \frac{3637}{578} a^{14} + \frac{12997}{578} a^{13} + \frac{2183}{289} a^{12} + \frac{18827}{578} a^{11} + \frac{3495}{289} a^{10} + \frac{33223}{578} a^{9} + \frac{7465}{289} a^{8} + \frac{24103}{289} a^{7} + \frac{14129}{578} a^{6} + \frac{21629}{289} a^{5} + \frac{2418}{289} a^{4} + \frac{10916}{289} a^{3} - \frac{789}{578} a^{2} + \frac{4451}{578} a - \frac{8}{289}$$,  $$\frac{1657}{578} a^{19} - \frac{33}{17} a^{18} + \frac{7171}{578} a^{17} - \frac{147}{17} a^{16} + \frac{7382}{289} a^{15} - \frac{627}{34} a^{14} + \frac{10178}{289} a^{13} - \frac{895}{34} a^{12} + \frac{15203}{289} a^{11} - \frac{666}{17} a^{10} + \frac{29717}{289} a^{9} - \frac{2541}{34} a^{8} + \frac{35743}{289} a^{7} - \frac{3183}{34} a^{6} + \frac{45377}{578} a^{5} - \frac{2243}{34} a^{4} + \frac{12483}{578} a^{3} - \frac{785}{34} a^{2} + \frac{1063}{578} a - \frac{121}{34}$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$176.861949817$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A non-solvable group of order 983040 The 155 conjugacy class representatives for t20n964 are not computed Character table for t20n964 is not computed

## Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

## Sibling fields

 Degree 20 siblings: data not computed Degree 40 siblings: data not computed

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ $16{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
magma: idealfactors := Factorization(p*Integers(K)); // get the data
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
gp: idealfactors = idealprimedec(K, p); \\ get the data
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5} 2.5.0.1x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.2$x^{10} - 5 x^{8} + 10 x^{6} - 2 x^{4} - 11 x^{2} + 39$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5} 1717.2.0.1x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2} 17.4.0.1x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4} 17.8.4.1x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
2297Data not computed